Re: Why are TI Calcs so inferior?
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In article <0mXwDl200bkv0Rd900@andrew.cmu.edu>, Nicholas P Konidaris Ii
<npk+@andrew.cmu.edu> wrote:
:Person who wrote the "Churning Butter Post" -- I'm afraid that your
:Calculus class IS Taught with a Ti-graphing-calculator because anyone
:who compares butter churning, which is a _Process_ to an abstract set of
:ideas, is being Taught A Process.
Well, some classes are, some aren't. I doubt you can make a very rigorous
high school calculus class without losing a lot of people. If people
aren't going into pure math, you could probably even argue that they don't
need to know the absolute proofs of all these things. After all, some of
this stuff isn't done really rigorously until real analysis.
:
:Everyone knows that:
:
:y(x) = x^n
:y'(x) = n * x^(n - 1)
:
:Something my ti-92 can symbolically show me. I'm sure you could even
:program it into a ti8x. Now, can any of you soft-minded HS students
:PROVE it? Qualitativly and Quantitativly?
Well, the proof of it with integers is actually rather complex, IIRC. You
have to write out what the sum of the k-th powers is. You could use
Bernoulli numbers, I suppose, but you can also come up with a recursion
from which you can get the leading term which is all you need. It
certainly is not trivial, however.
:
:OR is it simply a PROCESS which you have come to memorize?
Of course it is. It's certainly not intuitive. I've know the proof, but it
would take a few seconds to actually rederive the recursion above. I think
you can do the general proof by doing:
y = x^k => ln y = k ln x => (1 / y) dy/dx = k/x => dy/dx = y k / x = k x^(k -
1)
But, of course then you have to prove that d/dx (ln x) = 1/x. Probably
the best way way to go about it would be to define exp(x) to be the
function that has its derivative equal to itself. Then let y = exp(x) and
x = ln(y). Therefore dy/dx = exp(x). Therefore 1 = d(ln(y))/dx =
d(ln(y))/dy * dy/dx = exp(x) d(ln(y)) / dy => d(ln(y)) / dy = exp(-x) = 1
/ y. So you're done, if you've proved the chain rule and that exp(-x) = 1
/ exp(x), although the latter's fairly easy. Of course, you have to then
convince the students that exp(x) = e^x, etc. etc. etc. It takes a litte
work to do it rigorously. This would take away from trig-subsititutions,
integration by parts, polar coords, etc. Hell, they've taken the
multi-variable stuff off the AP already.
High Calculus is not a math class. It's a tool class so that people can
will have a sufficient introduction to do things where the theory might be
tangential to their works. While I'd love it if every engineer out there
took real analysis, there are probably better things they can be spending
their time on. It's not necessary for their job.
:
:Can you tell me what e is withouth thinking about it?
Well, the best way is to define it as above, IMO.
:
:How about the logarithm?
The inverse of e^x :)
:
:Sadly, a lot of _good_ students I know, who have done well on AP exams
:(4s and 5s), don't have this drilled into their head like they have 6*6.
: Why is that? Because they learned it ONCE in Algebra, or precalc, or
:whatever, and ever since then they have been using their calculators.
I'm not quite sure I see a problem here. I mean, really, how useful is it
to learn all the integration tricks in existance. I know that a while ago,
there were orders of magnitude more tricks taught in schools. The simple
fact is, though, it's a waste of time to integrate by hand. Do it once,
keep the theory in the back of your mind and let Mathematica do the ugly
algebra. I'm not sure this is a bad thing.
:
:Are graphing calculators nice? Sure, they can show a few things which
:would be very hard to show w/ just chalk. But they don't teach you
:calculus, they teach you how to plug equations in.
:
:We don't need any more hands-on-training in school folks. People have
:to learn to use their Brains. It's _FAR_ More usefull.
True. Supposedly geometry is the course where they teach you to be
rigorous. I absolutely hated geometry, however. People can learn to use
their brains, however, and still not have calculus taught rigorously.
Using your brain is a very wide idea and doing theoretical math is merely
one application.
Aaron (I don't _think_ I made a math mistake)
<pre>
--
Aaron Bergman -- abergman@minerva.cis.yale.edu
<http://pantheon.yale.edu/~abergman/>
Smoke a cigarette. Slit your throat. Same concept.
</pre>
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